Advertisement

GBI Method: A Powerful Technique to Study Drying of Complex Shape Solids

  • A. G. Barbosa de LimaEmail author
  • J. M. P. Q. Delgado
  • I. B. Santos
  • J. P. Silva Santos
  • E. S. Barbosa
  • C. Joaquina e Silva
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 48)

Abstract

This chapter briefly focuses on the theory and applications of drying process (heat and mass transfer) with particular reference to arbitrarily-shaped wet capillary-porous bodies. Here, a modeling based on the liquid diffusion theory and the mathematical formalism to obtain the exact solution of the governing equation via Galerkin-based integral method are presented. The model considers constant thermo-physical properties and convective boundary conditions at the surface of the solid. Applications have been done to different solids of revolution and wheat grain. Predicted results of the average moisture content, average temperature, and moisture content and temperature distributions within the porous solids are presented and discussed, and for particular situations they are compared with experimental drying data.

Keywords

Drying Analytical Wheat Prolate spheroid 

Notes

Acknowledgments

The authors would like to express their thanks to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brazil), and FINEP (Financiadora de Estudos e Projetos, Brazil) for supporting this work; to the authors of the references in this paper that helped in our understanding of this complex subject, and to the Editors by the opportunity given to present our research in this book.

References

  1. 1.
    Brooker, D.B., Bakker-Arkema, F.W., Hall, C.W.: Drying and storage of grains and oilseeds. AVI Book, New York (1992)Google Scholar
  2. 2.
    Brosnan, D.A., Robinson, G.C.: Introduction of drying of ceramics: with laboratory exercises The American Ceramic Society, Westerville (2003)Google Scholar
  3. 3.
    Strumillo, C., Kudra, T.: Drying: principles, science and design. Gordon and Breach Science Publishers, New York (1986)Google Scholar
  4. 4.
    Fortes, M.: A Non-equilibrium thermodynamics approach to transport phenomena in capillary-porous media with special reference to drying of grains and foods. Ph.D. thesis, Pardue University (1978)Google Scholar
  5. 5.
    Fortes, M., Okos, M.R.: Drying theories: their bases and limitations as applied to foods and grains. In: Mujumdar, A.S. (ed.) Advances in drying. Hemisphere Publishing Corporation, Washington (1980)Google Scholar
  6. 6.
    Crank, J.: The mathematics of diffusion. Oxford Science Publications, New York (1992)Google Scholar
  7. 7.
    Gebhart, B.: Heat conduction and mass diffusion. McGraw-Hill Inc, New York (1993)Google Scholar
  8. 8.
    Luikov, A.V.: Analytical heat diffusion theory. Academic Press, Inc., Ltd, London (1968)Google Scholar
  9. 9.
    Carslaw, H.S., Jaeger, J.C.: Conduction of heat in solids. Oxford University Press, New York (1959)Google Scholar
  10. 10.
    Elvira, C.: The diffusion process modelling in elliptic shaped bodies. In: Proceedings of the. 6th International Congress Engineering and Food, vol. 1, London (1990)Google Scholar
  11. 11.
    Haghighi, K., Irudayaraj, J., Stroshine, R.L., Sokhansanj, S.: Grain kernel drying simulation using the finite element method. Trans. ASAE 33(6), 1957–1965 (1990)CrossRefGoogle Scholar
  12. 12.
    Lu, R., Siebenmorgen, T.J.: Moisture diffusivity of long-grain in rice components. Trans. ASAE 35(6), 1955–1961 (1992)CrossRefGoogle Scholar
  13. 13.
    Sarker, N.N., Kunze, O.R., Stroubolis, T.: Finite element simulation of rough rice drying. Drying Technol. 12(4), 761–775 (1994)CrossRefGoogle Scholar
  14. 14.
    Lima, A.G.B.: Diffusion phenomena in solid prolate spheroid. Case study: Banana drying. Doctorate thesis. State University of Campinas, SP (1999)Google Scholar
  15. 15.
    Lima, A.G.B., Nebra, S.A.: Theoretical analysis of the diffusion process inside prolate spheroidal solids. Drying Technol. 18(1–2), 21–48 (2000)Google Scholar
  16. 16.
    Lima, A.G.B., Queiroz, M.R., Nebra, S.A.: Heat and mass transfer model including shrinkage applied to ellipsoids products: case study: drying of bananas. Develop. Chem. Eng. Mineral Process. 10(3–4), 281–304 (2002)Google Scholar
  17. 17.
    Lima, A.G.B., Nebra, S.A., Queiroz, M.R.: Simultaneous moisture transport and shrinkage during drying of solids with ellipsoidal configuration. Chem. Eng. J. 86(1–2), 85–93 (2002)CrossRefGoogle Scholar
  18. 18.
    Gastón, A.L., Abalone, R.M., Giner, S.A.: Wheat drying kinetics. Diffusivities for sphere and ellipsoid by finite elements. J. Food Eng. 52, 313–322 (2002)CrossRefGoogle Scholar
  19. 19.
    Carmo, J.E.F.: Diffusion Phenomena in oblate spheroidal solids: modelling and simulation. Master`s thesis, Federal University of Paraiba, Campina Grande (2000) (In Portuguese)Google Scholar
  20. 20.
    Carmo, J.E.F., Lima, A.G.B.: Drying of lentil including shrinkage: a numerical simulation. Drying Technol. 23, 1977–1992 (2005)CrossRefGoogle Scholar
  21. 21.
    Igathinathane, C., Chattopadhyay, P.K.: Surface area of general ellipsoid shaped food materials by simplified regression equation method. J. Food Eng. 46, 257–266 (2000)CrossRefGoogle Scholar
  22. 22.
    Carmo, J.E.F., Lima, A.G.B.: Mass transfer inside oblate spheroidal solids: modelling and simulation. Braz. J. Chem. Eng. 25(1), 19–26 (2008)CrossRefGoogle Scholar
  23. 23.
    Oliveira, V.A.B., Lima, W.C.P.B., Farias Neto, S.R., Lima, A.G.B.: Heat and mass diffusion and shrinkage in prolate spheroidal bodies based on non-equilibrium thermodynamics: a numerical investigation. J. Porous Media 14(7), 593–605 (2011)CrossRefGoogle Scholar
  24. 24.
    Carmo, J.E.F., Lima, A.G.B., Joaquina e Silva, C.: Continuous and intermittent drying (tempering) of oblate spheroidal bodies: modeling and simulation. Int. J. Food Eng. 8(3), 1–17 (2012) (Article 20)Google Scholar
  25. 25.
    Norminton, E.J., Blackwell, J.H.: Transient heat flow from constant temperature spheroids and the thin circular disk. Quart. J. Mech. Appl. Math XVII(1), 65–72 (1964)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Haji-Sheikh, A., Sparrow, E.M.: Transient heat conduction in a prolate spheroidal solid Trans. ASME. J. Heat Transf. 88(3), 331–333 (1966)CrossRefGoogle Scholar
  27. 27.
    Haji-Sheikh, A.: On solution of parabolic partial differential equations using Galerkin functions. In: Payne, F.R., Corduneanu, C.C., Haji-Sheikh, A., Huang, T. (eds.) Integral methods in science and engineering. Hemisphere Publishing Corporation, New York, USA (1986)Google Scholar
  28. 28.
    Haji-Sheikh, A., Lakshminarayanan, R.: Integral solution of diffusion equation: Part 2—boundary conditions of second and third kinds. J. Heat Transf. 109(3), 557–562 (1987)CrossRefGoogle Scholar
  29. 29.
    Oliveira, V.A.B., Lima, A.G.B., Joaquina e Silva, C.: Drying of wheat: a numerical study based on the non-equilibrium thermodynamics. Int. J. Food Eng. 8(3), 1–21 (2012) (Article 19)Google Scholar
  30. 30.
    Alassar, R.S.: Heat conduction from spheroids. J. Heat Transf. 121(2), 497–499 (1999)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Oliveira, V.A.B.: Diffusion in prolate spheroidal solids: an analytical solution, Master`s thesis, Federal University of Paraíba, Campina Grande (2001) (In Portuguese)Google Scholar
  32. 32.
    Oliveira, V.A.B., Lima, A.G.B.: Unsteady state mass diffusion in prolate spheroidal solids: an analytical solution. In: 2nd Inter-American Drying Conference. Boca Del Rio, Vera Cruz, Mexico (2001)Google Scholar
  33. 33.
    Oliveira, V.A.B., Lima, A.G.B.: Mass diffusion inside prolate spherical solids: an analytical solution. Braz. J. Agro-ind. Prod. 4(1), 41–50 (2002)Google Scholar
  34. 34.
    Farias, S.N.: Drying of spheroidal solids using the Galerkin method. Master’s thesis, Federal University of Paraíba, Campina Grande, Brazil (2002) (In Portuguese)Google Scholar
  35. 35.
    Lima, D.R., Farias, S.N., Lima, A.G.B.: Mass transport in spheroids using the Galerkin method. Braz. J. Chem. Eng. 21(1), 667–680 (2004)Google Scholar
  36. 36.
    Beck, J.V., Cole, K.D., Haji-Sheikh, A., Litkouhi, B.: Heat conduction using green’s functions. Hemispheric Publishing Corporation, New York (1992)Google Scholar
  37. 37.
    Hacihafizoglu, O., Cihan, A., Kahveci, K., Lima, A.G.B.: A liquid diffusion model for thin-layer drying of rough rice. European Food Res. Technol. 226(4), 787–793 (2008)CrossRefGoogle Scholar
  38. 38.
    Santos, I.B., Silva, L.P.L., and Lima, A.G.B.: Mass transport in solids with arbitrary shape via Galerkin-based integral method using convective boundary condition. In: 9th Argentinean Congress on Computational Mechanics and 31th Iberian-Latin-American Congress on Computational Methods in Engineering, vol. 1, pp. 2865–2881, Buenos Aires, Argentina (2010)Google Scholar
  39. 39.
    Silva, A.A., Santos, I.B., Lima, A.G.B.: Mass transport in solids with arbitrary shape via GBI method: an analytical study. In: 6th National Congress of Mechanical Engineering, vol. 1, Campina Grande, Brazil (2010)Google Scholar
  40. 40.
    Santos, I.B., Silva, L.P.L., Lima, A.G.B.: Mass transfer in irregularly-shaped solid: an exact solution using the galerkin-based integral method. Def. Diff. Forum 326–328, 199–204 (2012)CrossRefGoogle Scholar
  41. 41.
    Farias Neto, S.R., Farias, F.P.M., Delgado, J.M.P.Q., Lima, A.G.B., Cunha, A.L.: Cyclone: Their characteristics and drying technological applications. In: Delgado, J.M.P.Q. (ed.) Industrial and Technological Applications of Transport in Porous Materials, vol. 36, pp. 1–36 Springer, Heidelberg (2013)Google Scholar
  42. 42.
    Rosen, H.N.: Recent advances in the drying of solid wood. In: Mujumdar, A.S. (ed.) Advances in Drying, vol. 4, Hemisphere Publishing Corporation, Berlin (1987)Google Scholar
  43. 43.
    Kantorovich, L.V., Krylov, V.I.: Approximate methods of higher analysis. Advanced Calculus. Wiley, New York (1960)Google Scholar
  44. 44.
    Nascimento, J.J.S., Transient diffusion phenomenon in parallelepipeds solids. Case studied: Drying of ceramic materials. Ph.D. thesis, Federal University of Paraíba, João Pessoa (2002) (In Portuguese)Google Scholar
  45. 45.
    Fortes, M., Okos, M.R., Barret Jr, J.R.: Heat and mass transfer analysis of intra-kernel wheat drying and rewetting. J. Agric. Eng. Res. 26, 109–125 (1981)CrossRefGoogle Scholar
  46. 46.
    Fioreze, R.: The intermittent drying agricultural crops with particular reference to energy requirements. Doctorate thesis Cranfield Institute of Technology, University of Cranfield, UK (1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • A. G. Barbosa de Lima
    • 1
    Email author
  • J. M. P. Q. Delgado
    • 2
  • I. B. Santos
    • 3
  • J. P. Silva Santos
    • 1
  • E. S. Barbosa
    • 1
  • C. Joaquina e Silva
    • 1
  1. 1.Department of Mechanical EngineeringFederal University of Campina GrandeCampina GrandeBrazil
  2. 2.LFC—Building Physics Laboratory, Civil Engineering Department, Faculty of EngineeringUniversity of PortoPortoPortugal
  3. 3.Department of PhysicsState University of ParaibaCampina GrandeBrazil

Personalised recommendations